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Central limit theorems for law-invariant coherent risk measures

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  • Belomestny, Denis
  • Krätschmer, Volker

Abstract

In this paper we study the asymptotic properties of the canonical plug-in estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent identically distributed data and then extend it to the case of weakly dependent ones. Finally, a number of illustrating examples is presented.

Suggested Citation

  • Belomestny, Denis & Krätschmer, Volker, 2010. "Central limit theorems for law-invariant coherent risk measures," SFB 649 Discussion Papers 2010-052, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2010-052
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    File URL: https://www.econstor.eu/bitstream/10419/56647/1/641641230.pdf
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    References listed on IDEAS

    as
    1. Krätschmer, Volker, 2007. "On {sigma}-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model," SFB 649 Discussion Papers 2007-010, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
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    Cited by:

    1. Basteck, Christian & Daniëls, Tijmen R., 2010. "Every symmetric 3 x 3 global game of strategic complementarities is noise independent," SFB 649 Discussion Papers 2010-061, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    2. repec:hum:wpaper:sfb649dp2010-061 is not listed on IDEAS
    3. repec:hum:wpaper:sfb649dp2010-059 is not listed on IDEAS
    4. Schulze, Franziska, 2010. "Spatial dependencies in German matching functions," SFB 649 Discussion Papers 2010-054, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Niushan Gao & Denny Leung & Cosimo Munari & Foivos Xanthos, 2018. "Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Finance and Stochastics, Springer, vol. 22(2), pages 395-415, April.
    6. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.
    7. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
    8. repec:hum:wpaper:sfb649dp2010-056 is not listed on IDEAS
    9. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    10. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    11. Wiebach, Nicole & Hildebrandt, Lutz, 2010. "Context effects as customer reaction on delisting of brands," SFB 649 Discussion Papers 2010-056, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    12. Lauer Alexandra & Zähle Henryk, 2016. "Nonparametric estimation of risk measures of collective risks," Statistics & Risk Modeling, De Gruyter, vol. 32(2), pages 89-102, March.
    13. Kamal Boukhetala & Jean-François Dupuy, 2014. "Colloque International Book of Abstracts Edité par K. Boukhetala et J.F Dupuy," Post-Print hal-01086342, HAL.
    14. repec:hum:wpaper:sfb649dp2010-055 is not listed on IDEAS
    15. repec:hum:wpaper:sfb649dp2010-054 is not listed on IDEAS
    16. Nikolaus Hautsch & Peter Malec & Melanie Schienle, 2014. "Capturing the Zero: A New Class of Zero-Augmented Distributions and Multiplicative Error Processes," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 89-121.
    17. Mammen, Enno & Rothe, Christoph & Schienle, Melanie, 2010. "Nonparametric regression with nonparametrically generated covariates," SFB 649 Discussion Papers 2010-059, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    18. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2014. "Quasi-Hadamard differentiability of general risk functionals and its application," Papers 1401.3167, arXiv.org, revised Feb 2015.

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    More about this item

    Keywords

    law-invariant coherent risk measures; canonical plug-in estimates; functional central limit theorems; weak dependence;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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